Q.Reaction time: When a situation demands our immediate action, it takes some time before we really respond. Reaction time is the time a person takes to observe, think and act. For example, if a person is driving and suddenly a boy appears on the road, then the time elapsed before he slams the brakes of the car is the reaction time. Reaction time depends on complexity of the situation and on an individual. You can measure your reaction time by a simple experiment. Take a ruler and ask your friend to drop it vertically through the gap between your thumb and forefinger. After you catch it, find the distance $d$ travelled by the ruler. In a particular case, $d$ was found to be $21.0\ \text{cm}$. Estimate reaction time.

The key idea is that the ruler is in free fall under gravity during the reaction time, so the distance it falls relates directly to the time elapsed.

1. The ruler starts from rest ($u = 0$) and falls with acceleration $g = 9.8\ \text{m/s}^2$ (taking standard value).
2. Using the equation of motion: $d = \frac{1}{2} g t^2$, where $d = 21.0\ \text{cm} = 0.210\ \text{m}$. …

The idea behind this experiment is beautifully simple. When your friend releases the ruler, it begins to fall freely under gravity. Your job is to catch it as fast as you can after you see it drop. The distance the ruler falls before you catch it is directly related to how long it took your brain and muscles to react — that’s your reaction time. Because the ruler starts from rest and accelerates uniformly (ignoring air resistance), we can use the equations of motion for constant acceleration.

The key assumption is that the ruler’s motion is pure free fall, so its acceleration is $g = 9.8\ \text{m/s}^2$ downward. The distance $d$ it falls in time $t$ is given by $d = \frac12 g t^2$, since initial velocity is zero. We just need to solve for $t$.

1. Convert distance to metres. The given $d = 21.0\ \text{cm}$ must be in SI units to match $g$ in $\text{m/s}^2$.

   $d = 21.0\ \text{cm} = 0.210\ \text{m}$.

2. Write the free-fall equation.

3. Solve for $t$.

4. Substitute the numbers.

Compute the fraction:

Then take the square root:

$t = \sqrt{0.042857} \approx 0.207\ \text{s}$ …

Concept: Deriving the Free-Fall Distance Formula from the v-t Graph, Then Inverting It

Method: Build the $d = \tfrac{1}{2}gt^2$ Relation from Its Graph, Rather Than Recalling It

Instead of directly quoting $d=\tfrac{1}{2}gt^2$ and solving for $t$, this method derives that formula from scratch as the area under the ruler's $v$-$t$ graph, which also makes clear why the exponent on $t$ is exactly 2.

Steps

1. Model the ruler's fall. Released from rest, the ruler undergoes free fall: $v(t) = 0 + gt = gt$ — a straight line through the origin on a $v$-$t$ graph, with slope $g$.

2. Distance fallen = area under this line, from $0$ to $t$. That region is a triangle with base $t$ and height $v(t) = gt$:

This is the formula used by the direct method — here it's derived, not recalled.

3. Convert the measured distance to SI units:

4. Invert the relation to solve for $t$:

5. Substitute $g = 9.8\ \text{m s}^{-2}$: …